The Analytic Solution Of The Schr(?)dinger Equation And Dirac Equation For The Superposition Potential Of The Power Functions

Analytic solution and approximate solution for solving the stationary state Schr?ddinger equation are presented . And the radial Schr?dinger equation with the three power and four inverse power potential function is studied in quantum mechanics. Generally speaking, the potential under the conditions of superposition are not the analytical solution is given . But if the power coupling function of the close relationship between the cases, there may be analytic solution and energy spectrum equations.In paper,according to quantum system wave function must meet the standard conditions of single value, continuous and limited, First obtained radial coordinates r→0and the r→∞analytical solution, Then use neighborhood near the singular point of the series and the method of asymptotic solution obtained by combining indicators s power function, and the coefficient of the constraint, an analytic solution and energy level of the superposition potential ( )been obtained and some conclusions are presented, and using MATLAB software distribution wave function graphics.A new ring-shaped non-harmonic oscillator potential ( )is proposed in paper.Firstly by using the usual method of variable separation, the normalized angle wave function and normalized radial wave function are obtained under the condition of equal scalar and vector potentials. The normalized angle wave function is expressed in terms of the universal associated-Legendre polynomial, normalized radial wave function and energy spectrum equations are expressed in terms of the confluent hypergeometric function. Secondly,exact bound states of Dirac equation with scalar and vector potentials are solved.Finally, properties of the system relate to three quantum numbers( nr , m, s )and parameters( K , A,β,γ)of potential.